Regularization in Hilbert space under unbounded ... - TU Chemnitz

IOP PUBLISHING

INVERSE PROBLEMS

Inverse Problems 25 (2009) 115013 (15pp)

doi:10.1088/0266-5611/25/11/115013

Regularization in Hilbert space under unbounded operators and general source conditions* Bernd Hofmann1 , Peter Mathé2 and Heinrich von Weizsäcker3 1

Department of Mathematics, Chemnitz University of Technology, 09107 Chemnitz, Germany Weierstraß Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany 3 Department of Mathematics, Technical University of Kaiserslautern, Erwin-Schrödinger-Straße, 67655 Kaiserslautern, Germany 2

E-mail: [email protected], [email protected] and [email protected]

Received 1 July 2009, in final form 22 September 2009 Published 29 October 2009 Online at stacks.iop.org/IP/25/115013 Abstract The authors study ill-posed equations with unbounded operators in Hilbert space. This setup has important applications, but only a few theoretical studies are available. First, the question is addressed and answered whether every element satisfies some general source condition with respect to a given self-adjoint unbounded operator. This generalizes a previous result from Mathé and Hofmann (2008 Inverse Problems 24 015009). The analysis then proceeds to error bounds for regularization, emphasizing some specific points for regularization under unbounded operators. The study finally reviews two examples within the light of the present study, as these are fractional differentiation and some Cauchy problems for the Helmholtz equation, both studied previously and in more detail by U Tautenhahn and co-authors.

1. Introduction and main result There are many papers that outline the regularization theory of ill-posed operator equations (with given right-hand side y and unknown solution x) A x = y,

(1)

where the forward operator A is an injective linear and bounded mapping between two Hilbert spaces X and Y with inner products ·, · and associated norms · . The ill-posedness of the problem comes from the fact that the range R(A) is a non-closed subset of Y, i.e., R(A) = R(A).

(2)

* To Eberhard Schock on the occasion of his 70th birthday. 0266-5611/09/115013+15$30.00 © 2009 IOP Publishing Ltd Printed in the UK

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Inverse Problems 25 (2009) 115013

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Following Nashed (see [8]) we have to distinguish ill-posedness of type I, where condition (2) is complemented by the fact that the range R(A) contains a closed infinite dimensional subspace, and ill-posedness of type II, where A with infinite-dimensional range is compact. In this paper, we focus on more general ill-posed equations (1) with a closed, densely defined linear operator A : D(A) ⊆ X −→ Y, with domain D(A), i.e., D(A) is a dense linear subspace of X, and convergences xn → x0 in X with xn ∈ D(A) (n ∈ N) and Axn → y in Y imply x0 ∈ D(A) and Ax0 = y0 . This includes the cases that either the operator A is bounded (with D(A) = X) but not necessarily compact (see also [4]) or that A with a proper subset D(A) of X is unbounded. One can find in the literature only a few studies which treat regularization theory under unbounded operators. The monograph [15] discusses inverse problems under unbounded operators, but only marginally and without details. We also mention [10], where Tikhonov regularization is considered. Therefore, we feel that it makes sense to highlight some important features of this more general setup. Special care is required to apply the usual functional calculus from regularization theory, see [1, 7]. There are interesting applications which lead to the problem of regularization under unbounded operators, and we review some of them in section 4. Specifically, inverse problems under non-compact and unbounded operators cover cases beyond integral equations as shown in section 4.1. However, also for initial and boundary value problems of partial differential equations such situations may occur. An example is the Cauchy problem for the Helmholtz equation, which is presented in section 4.2. Smoothness is often measured in terms of source conditions. The most general setup assumes that a solution to (1) belongs to the range of some index function applied to the operator A∗ A, and we refer to section 2 for a formal introduction. The main objective of this study is to prove the existence of such source condition whenever the underlying operator A from (1) is a closed, densely defined and injective operator. In that case the operators A∗ A and AA∗ are non-negative self-adjoint operators. Theorem 1. Let X be a Hilbert space and let H : D(H ) ⊆ X → X be an injective, non-negative self-adjoint linear operator. Then the following results hold true. (a) For every element x ∈ X and ε > 0 there is a bounded index function ψ : (0, ∞) → (0, ∞) such that the general source condition x = ψ(H )w

with w ∈ X

and

w (1 + ε)x

is satisfied, and hence x ∈ R(ψ(H )). (b) If x ∈ R(ψ(H )) for some unbounded index function ψ, then x ∈ R(ψ0 (H )) for every bounded index function ψ0 which coincides with ψ on (0, t0 ] for some t0 > 0. Below, we shall agree to call a self-adjoint operator positive if it is injective and nonnegative. We shall first develop some calculus for domains and ranges for positive self-adjoint operators, and then prove with theorem 1 the main result. In particular, we highlight that smoothness given in terms of Sobolev–Hilbert spaces can be translated to some source condition, a fact which is easy to establish but may not be of common knowledge. After that we are going to discuss several topics of regularization theory under unbounded operators, which differ in some aspects from the classical one. We conclude with two examples. 2


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2. Calculus for general source conditions under unbounded operators Source conditions use index functions and these must be defined on the spectrum σ (H ) of some positive self-adjoint operator H. If this operator is unbounded, then it is reasonable to assume that the corresponding function is defined on R+ . Hence, we call a positive function ψ : (0, ∞) → (0, ∞) an index function if it is an increasing (non-decreasing) and continuous function with limt→0 ψ(t) = 0. We consider both bounded and unbounded index functions, i.e. limt→∞ ψ(t) can be finite or +∞. In [6] the authors have shown the existence of an index function ψ for every element x † ∈ X in the Hilbert space X such that a general source condition x † = ψ(H ) with source element w ∈ X holds whenever H is a positive and compact self-adjoint linear operator, which implies that H is bounded. Theorem 1 extends the theorem from [6] to the missing cases that either H is bounded but non-compact or that H is unbounded. When measuring smoothness with respect to an operator equation (1), we let H := A∗ A. Under the assumed closedness of A the adjoint operator A∗ is densely defined, and both operators A∗ A and AA∗ are self-adjoint, see [11, chapter VIII]. We agree to say that the solution x † satisfies a source condition if there are a (bounded or unbounded) index function ψ and a source element w ∈ D(ψ(A∗ A)) such that x † = ψ(A∗ A)w.

(3) †

∗

Plainly this is a reformulation of x ∈ R(ψ(A A)). In case of doubt we say in extenso that x † satisfies a source condition given by the index function ψ with source element w. 2.1. Domain and range relations for index functions Here we develop some calculus for operators given by index functions of some positive selfadjoint operator H. The following lemma establishes that given such an operator H, and some (bounded or unbounded) index function ψ, the range of ψ(H ) is determined by the behavior of ψ near zero. We shall use the following calculus for ranges and domains of operator functions. For this purpose we restrict our attention to unbounded index functions when studying domains of operators. If H is some unbounded positive self-adjoint operator H mapping in X, and if the index function ψ is unbounded, then the associated operator ψ(H ) is unbounded, positive ˜ given by ψ(t) ˜ and self-adjoint. Furthermore, the mapping ˜ : ψ → ψ, := 1/ψ(1/t), t > 0,4 maps unbounded index functions to unbounded index functions, and it holds true that −1 ˜ D(ψ(H )) = R((1/ψ)(H )) = R(ψ(H )).

(4)

We shall also use the following simple facts. Lemma 1. Let H be a positive self-adjoint operator. If two index functions ψ and ψ obey 0 < ψ (t) ψ(t) < ∞ for t > 0, then we have R(ψ (H )) ⊆ R(ψ(H ))

and

D(ψ(H )) ⊆ D(ψ (H )).

Proof. The range inclusion is easy to see. By the assumed ordering the operator ψψ (H ) is bounded. If for some v ∈ X we have that x † = ψ (H )v then w := ψψ (H )v is in the domain of ψ(H ), and we have the identity x † = ψ (H )v = ψ(H ) ψψ (H )v = ψ(H )w. The domain inclusion is trivial if ψ is bounded. Otherwise both index functions are unbounded, and the 4

For monomials ψp (t) := t p this mapping is identical as we have that ψ˜ p = ψp . 3


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domain inclusion is a consequence of the range inclusion. Indeed, if ψ ψ then this yields that ψ˜ ψ˜ , and thus −1 ˜ D(ψ(H )) = R(ψ(H )) ⊆ R(ψ˜ (H −1 )) = D(ψ (H )),

which completes the proof.

Lemma 2. Let H : D(H ) ⊆ X → X be a positive self-adjoint operator and suppose that ψ1 , ψ2 are two index functions. If both index functions coincide on (0, t0 ], for some t0 > 0, then R(ψ1 (H )) = R(ψ2 (H )). Proof. Given t0 > 0, we can assign to any index function ψ the bounded index function ψ(t), 0 < t t0 ψ0 (t) := (5) ψ(t0 ), t > t0 . We claim that R(ψ(H )) = R(ψ0 (H )), from which the assertion of the lemma is a consequence. The inclusion R(ψ0 (H )) ⊆ R(ψ(H )) follows from lemma 1 since ψ0 ψ. The other inclusion is a consequence of the decomposition ψ ψ ψ (t) = (t)χ0 0 there is a sequence σ0 = 1 σ1 σ2 · · · > 0 such that limk→∞ σk = 0, and ∞ gχk 2 (1 + ε)2 g2 . 2 σ k k=0

This follows, for example as in the proof of the theorem in [6], from [9, section 8.6.4]. On that basis we define a continuous piecewise linear function ψ which takes the values σk at the grid point t = 2−k and ψ(t) = 1 for t 1. Then ψ is an index function in our sense. We let g˜ := g/ψ(f ) and claim that this function belongs to L2 (, A, μ) and satisfies the norm bound. By construction we have that ∞ |g(t)|2 |g(t)|2 gχk 2 ˜ 2= g dμ dμ = (1 + ε)2 g2 , 2 2 2 ˜ σ ψ(f (t)) ψ(f (t)) k k=0

and the proof is complete.

Remark 2. We should mention the following consequence. Suppose that := (0, ∞), equipped with the Borel σ -algebra, and some measure μ, and let f (s) = s, s > 0. Then the assertion of lemma 4 may be rephrased as follows: for each square integrable function g there is an index function ψ such that g/ψ is still square integrable. To each real-valued measurable function f on a measure space (, A, μ), we may assign a multiplication operator Mf in L2 (, A, μ), given by Mf h := f · h.

(7)

Its domain of definition is given as D(Mf ) = {h ∈ L2 (, A, μ) : f · h ∈ L2 (, A, μ)}.

(8)

In complex Hilbert space X, each self-adjoint operator, say H : D(H ) ⊆ X → X is unitarily invariant to a multiplication operator, i.e., there are a measure space (, A, μ), a measurable function f and a unitary operator U : X → L2 (, A, μ) such that x ∈ D(H ) ⇐⇒ f · U x ∈ L2 (, A, μ)

(9)

and U H U ∗h = f · h

whenever

h ∈ D(Mf ).

(10)

Moreover, for any bounded measurable function ψ it holds that ψ(H ) = U ∗ ψ(Mf )U = U ∗ Mψ(f ) U.

(11)

We refer to [11, chapter VIII] for details. Remark 3. Actually, in [11], the proof of the spectral theorem is presented for separable Hilbert spaces, and it is shown that in this case the representing measure μ can be chosen 5


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finite. However, the theorem also holds for non-separable Hilbert spaces, but the structure of the measure space (, A, μ) may be more complicated. Most proofs of the spectral theorem for unbounded operators use the Caley transform, and are thus formulated in complex Hilbert space. Other proofs reduce the spectral theorem for unbounded operators to the bounded case by using a suitable transformation, see [2, section 30]. In this case the argument works for both, real and complex Hilbert spaces. Remark 4. The relation between positive self-adjoint operators and multiplication operators, as presented above sheds some new light on the assertions of lemmas 1–3. Indeed, it holds x ∈ D(ψ(Mf )) if x · ψ(f ) ∈ L2 (, A, μ), see (8), and consequently that x ∈ R(ψ(Mf )) if x/ψ(f ) ∈ L2 (, A, μ), giving rise to alternative proofs of the lemmas. As the above discussion reveals, multiplication operators are representative for (unbounded) self-adjoint operators, and properties of the operator H can be seen as properties of the representing function f in Mf. Specifically we call R(f ) := σ (Mf ) \ {0} = σ (H ) \ {0} the essential range of the function f . We refer to [11, chapter VII.2] for more details. Ill-posed problems are characterized by operators H with a non-closed range R(H ), where zero is an accumulation point of the set R(f ). Also, a positive operator H is unbounded if and only if +∞ is an accumulation point of R(f ). Example 1 (Sobolev–Hilbert spaces). The above approach is taken in the definition of Sobolev–Hilbert spaces on R. We let f (s) := (1 + s 2 )1/2 , s ∈ R. For any p > 0 we assign the positive self-adjoint multiplication operator Mf p with corresponding domain. The Fourier transformation F constitutes an isometry on L2 (R), and we then let Gp := F Mf p F −1 in L2 (R) with domain H p (R) := {x ∈ L2 (R) :

xp := F Mf p F −1 xL2 (R) < ∞}.

We refer to [14, 2.3.3] for a general outline. The operator Gp is clearly semi-bounded, and hence we may apply the reasoning from p (R) belongs remark 1: an element −1x ∈ H p to the Sobolev space exactly if it satisfies a source . This will be relevant for the example of fractional condition x ∈ R Gp = R G−1 1 differentiation, studied in section 4.1. Note that the function f representing the operator Gp has the essential range R(f ) = [1, ∞), indicating that Gp possesses a closed range. In the following lemma, we shall assume that the measure μ is finite, which is typical for separable Hilbert spaces X, see remark 3. The result extends to the non-separable case with a technical modification of the proof. Lemma 5. If the non-negative self-adjoint operator H : D(H ) ⊆ X → X is injective, then the measurable function f which represents the corresponding multiplication operator can be chosen to be strictly positive. Proof. This follows from the following identity for functions 0 = h ∈ D(Mf ): f |h|2 dμ = Mf h, h = U H U ∗ h, h = H U ∗ h, U ∗ h > 0.

Assume now that f 0 on a set S of positive measure. Making S smaller if necessary we may also assume that f is bounded on S. Since μ is finite the function h = χS then is in D(Mf )\{0} which yields a contradiction to the estimate above. Hence f > 0 μ-a.e. and U ∗ h = 0. Hence 6


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f would be a function with f |h|2 dμ = 0. Therefore f > 0 μ-a.e. and we can change f to be positive on the remaining null set. We are now in a position to prove the main result. Proof of theorem 1. Under the assumptions on H, by using lemma 5, the representing multiplication operator Mf possesses a positive function f . If x ∈ X then g := U x ∈ L2 (, A, μ), and we can apply lemma 4 to find a bounded index function ψ such that g˜ := g/ψ(f ) ∈ L2 (, A, μ). By using (11) this means that x = U ∗ g = U ∗ Mψ(f ) g˜ = U ∗ ψ(Mf )U U ∗ g˜ = ψ(H )w, where we defined w := U ∗ g˜ ∈ X. This proves assertion (a). On the other hand, for proving (b) let x ∈ R(ψ(H )) for some unbounded index function. Then we can replace this by any bounded one as done in lemma 2, and the proof is complete. We draw some consequences of theorem 1 in the following. Corollary 1. Under the assumptions of theorem 1, the following holds true. (a) If x ∈ R(ψ(H )) for some index function ψ, then there is an index function ψ1 such that x ∈ R([ψ · ψ1 ](H )). (b) If x ∈ D(ψ(H )) for some unbounded index function ψ, then there is an unbounded index function ψ1 such that x ∈ D([ψ · ψ1 ](H )). Proof. To prove (a) we apply theorem 1 once more to w ∈ X in order to find an index function ψ1 with w = ψ1 (H )v, which proves the result. To prove (b) we assume without loss of generality that the operator H is unbounded. Then we use the identity (4) to infer that −1 ˜ )). By item (a) there is an index function ψ1 such that x ∈ R([ψ˜ · ψ1 ](H −1 )). x ∈ R(ψ(H We note that the function ψ˜ · ψ1 constitutes an unbounded index function, and applying the identity (4) once more we obtain that x ∈ D([ψ · ψ˜ 1 ](H )). The function ψ˜ 1 may not be an index function as it may be bounded away from zero. However, in the light of lemma 3 we may replace ψ˜ 1 by any unbounded index function sharing the same behavior near infinity, which concludes the proof of the corollary. Remark 5. Assertion (a) of the corollary strengthens the assertion of theorem 1, and it emphasizes that there is no maximal smoothness, a fact which has various consequences in regularization theory, a topic which will not be discussed here. 3. Regularization under unbounded operators As mentioned in section 1, there are only a few papers which study linear equations (1) for unbounded operator A : D(A) ⊆ X → Y with x ∈ D(A), and we recall the study [10] in which Tikhonov regularization is considered. There, the solution representation for Tikhonov regularization is derived from minimizing the functional Jα (x, y δ ) := Ax − y δ 2 + αx2 ,

α > 0.

Theorem 2 in [10] establishes that there is a unique solution, say xαδ , to this problem, and this solution has the representation xαδ := A∗ (AA∗ + αI )−1 y δ

(12)

for any data y ∈ Y , where we assume y − y δ. We shall extend this to more general regularization, and we recall the notion of linear regularization in Hilbert space, see e.g. [5, definition 2.2]. δ

δ

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Definition 1. A family of functions gα : (0, ∞) → R, 0 < α α, ¯ of bounded measurable functions is called regularization if they are piecewise continuous in α and the following properties hold, where rα (t) := 1 − tgα (t), t > 0, denotes the residual function. (a) For each t > 0 there is convergence |rα (t)| → 0 as α → 0. γ1 for all 0 < √ α α. ¯ (b) There is a constant γ1 such that |rα (t)| √ ¯ (c) There is a constant γ∗ such that supt>0 t|gα (t)| γ∗ / α, for 0 < α α. Remark 6. For Tikhonov regularization as in (12), the family is given by gα (t) := 1/(t + α), and it is easily seen to satisfy the properties of a regularization. In analogy to (12), given some general regularization, and data y δ ∈ Y we let xαδ := A∗ gα (AA∗ )y δ

(13)

be the approximate solution to (1) by using the parameter α. Our goal is first to see why (13) is correctly defined in the case of an unbounded operator A, and then to derive error bounds under smoothness in terms of general source conditions x † = ψ(H )w, with H = A∗ A, for the solution x † . 3.1. Auxiliary analysis In the light of definition 1(c) we introduce the following space W of bounded (Borelmeasurable) functions on R+ as √ + + W := g : R → R bounded and gW := sup s|g(s)| < ∞ . (14) s∈R+

The calculus for general regularization schemes is based on the following. Proposition 1. Let A : D(A) ⊂ X → Y be a densely defined and closed operator, and let g be any bounded measurable function in R+ . If y ∈ D(A∗ ) then also g(AA∗ )y ∈ D(A∗ ), and it holds that A∗ g(AA∗ )y = g(A∗ A)A∗ y.

(15)

In addition, for all functions in W the operator defined in (15) extends to a bounded operator from Y → X, with norm bounded by gW . (Analogous statements hold for the equation Ag(A∗ A) = g(AA∗ )A.) Proof. We shall use the polarization identity for closed densely defined operators, see e.g. [11, theorem VIII.32], i.e., there is a partial isometry U (on ker(A)⊥ onto the closure of R(A)) such that A = U |A|, where |A| := (A∗ A)1/2 . Therefore, it holds that AA∗ = U |A|2 U ∗ , and for any y ∈ D(A∗ ) we have that g(A∗ A)A∗ y = g(|A|2 )|A|U ∗ y = |A|g(|A|2 )U ∗ y = A∗ Ug(|A|2 )U ∗ y = A∗ g(AA∗ )y. Thus, we have that g(AA∗ )y ∈ D(A∗ ) and the identity (15) holds. Finally, if g ∈ W then we assign to a given y ∈ D(A∗ ) the element z := g(A∗ A)y ∈ D(A∗ ). We have that A∗ z2 = A∗ z, A∗ z = AA∗ z, z = (AA∗ )1/2 z2 g2W y2 , ∗

(16)

∗

and the operator from A g(A A) extends to a bounded operator on all of Y with norm bound gW . The proof is complete. This allows for the following consequences with regard to error bounds of regularization. 8


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Corollary 2. Let A : D(A) ⊂ X → Y be a closed, densely defined operator, and let ¯ constitute a regularization. Then the following assertions hold true: gα , 0 < α α, (a) The identity A∗ gα (AA∗ )A = gα (A∗ A)A∗ A holds, and the self-adjoint operator rα (A∗ A) extends to a bounded operator in X with norm less than or equal to γ1 . (b) For each y ∈ Y the element gα (AA∗ )y belongs to the domain of A∗ and y A∗ gα (AA∗ )y γ∗ √ . α Both assertions are consequences of proposition 1 and we omit the proof. The norm bound in (b) √ follows as in the proof of (16), and spectral calculus yields that in this case gα W γ∗ / α. Assertion (b) confirms that xαδ from (13) is well defined for any data y δ ∈ Y . Correspondingly to (13), we let xα := A∗ gα (AA∗ )Ax † ,

(17)

†

for the solution x ∈ D(A), which is also well defined by corollary 2. Error bounds are obtained from † x − x δ x † − xα + xα − x δ , (18) α α and we treat both summands, separately. An estimate for the second term in the right-hand side of (18) expresses the noise propagation, and it holds Lemma 6. For each α > 0 we have that xα − x δ γ∗ √δ . α α Proof. We rewrite xα − x δ = A∗ gα (AA∗ )(Ax † − y δ ) δA∗ gα (AA∗ ). α

Corollary 2(b) allows us to complete the proof.

The first term x † − xα , which is called bias, requires smoothness of the solution x † , and by theorem 1 we know that there exists a bounded index function ψ such that x † satisfies a source condition (3). Lemma 7. Suppose that the solution satisfies a source condition (3) given by the index function ψ and with a source element w ∈ D(ψ(H )). Then for each 0 < α α¯ we have that x † − xα w

sup

s∈σ (A∗ A)

|rα (s)|ψ(s).

Proof. We rewrite x † − xα = x † − A∗ gα (A∗ A)Ax † . Using corollary 2(a) we can continue with x † − xα = (I − A∗ gα (AA∗ )A)x † = rα (A∗ A)ψ(A∗ A)w wrα (A∗ A)ψ(A∗ A). We exploit functional calculus from (11) for the self-adjoint operator A∗ A to conclude that rα (A∗ A)ψ(A∗ A) sup |rα (f (t))|ψ(f (t)) t∈

by substituting s := f (t) > 0.

sup

s∈σ (A∗ A)

|rα (s)|ψ(s),

(19) 9


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3.2. Qualification of regularization under unbounded operators Bounds for the bias are important for understanding regularization, and the related concept is called qualification. For the convenience of the reader we recall this notion, see [5, definition 2.6]. The goal is to bound the bias at the solution x † = ψ(A∗ A)w in terms of the involved index function ψ, as this was done in (19), at least for α > 0 small. This leads to the following definition. Definition 2. An index function ψ is a qualification of the regularization gα , if there are a constant γ := γψ < ∞ and a value 0 < α¯ < ∞ such that sup

s∈σ (A∗ A)

|rα (s)|ψ(s) γ ψ(α),

0 < α α. ¯

(20)

In the case that the spectrum σ (A∗ A) is bounded, i.e., for bounded operators A, it is well known that the qualification is determined in a neighborhood of zero. Moreover, if one index function is a qualification then this is also valid for all index functions which coincide with it in a neighborhood of zero. However, such assertion does not hold under unbounded operators. It extends only if we impose some growth condition (GC). The following result is a generalization of [5, proposition 2.7] to general regularization and under unbounded operators. Proposition 2. Let gα be a regularization with some known qualification ϕ. An index function ψ is a qualification of gα if there is s0 > 0 for which the function s → ψ(s)/ϕ(s), 0 < s s0 , is non-increasing and moreover the growth condition C :=

sup s>s0 ,

s∈σ (A∗ A)

ψ(s) s¯ with s ∈ σ (A∗ A). In the first case by using the monotonicity we have that |rα (s)|ψ(s) γ1 ψ(α), with γ1 from definition 1(b). For α < s s¯ using the assumed monotonicity of the quotient we have ψ(s) ψ(α) γϕ ϕ(α) γϕ ψ(α). ϕ(s) ϕ(α) For the remaining case α < s¯ < s we reduce the problem to the known qualification of the function ϕ. We bound ψ(s) ϕ(α) ψ(s) γϕ ψ(α) |rα (s)|ψ(s) = |rα (s)|ϕ(s) ϕ(s) ψ(α) ϕ(s) ϕ(s0 ) ψ(s) ϕ(s0 ) γϕ ψ(α) γϕ C ψ(α), ψ(s0 ) ϕ(s) ψ(s0 )

ϕ(s0 ) . yielding the qualification bound with γψ := max γϕ , γ1 , γϕ C ψ(s 0) |rα (s)|ψ(s) = |rα (s)|ϕ(s)

Remark 7. Condition (GC) is a growth constraint at the infinite part of the spectrum. If this holds for some s0 > 0 then it holds for every s0 > 0. There are two important cases where the constant C in (GC) is finite. First, if the operator A is bounded (in which case C = ψ(A∗ A)/ϕ(s0 )), or in the case that the index function ψ is uniformly bounded, say by C · ψ(s0 ). Therefore, this additional assumption (GC) cannot be seen when treating qualification under bounded operators. 10


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If the monotonicity assumption on ψ(s)/ϕ(s) in proposition 2 holds globally, then (GC) is automatically fulfilled with C = ψ(s0 )/ϕ(s0 ). The monotonicity assumption of the above proposition represents low smoothness. We accompany the above result with the case of high smoothness. It yields the best to expect even under highest possible smoothness. The proof is similar to the above one and we omit it. Proposition 3. Let gα be a regularization with known qualification ϕ. Suppose that the index function ψ obeys (GC). If there is s0 > 0 such that the function ψ(s)/ϕ(s) is non-decreasing for all 0 < s s0 , then there is a constant 1 C < ∞ such that sup

s∈σ (A∗ A)

|rα (s)|ψ(s) Cϕ(α)

for all

0 < α α. ¯

The notion of qualification uses global properties of the index functions on all of σ (A∗ A). It thus may happen that for two functions which coincide on some interval (0, s0 ] one is a qualification of the regularization and the other is not. This is a paradox which cannot be observed for bounded operators and it is artificial. In fact, by using theorem 1(b) based on lemma 2 we can replace any unbounded index function ψ by a bounded one ψ0 , which equals the original one on some initial segment (0, s0 ]. For the latter ψ0 , the proposition applies and thus yields a convergence rate for the bias. Taking this into account we summarize the above analysis as Proposition 4. Let gα be a regularization with known qualification ϕ. Suppose that the solution satisfies a source condition (3) given by the index function ψ with source element w ∈ X. (a) If there is s0 > 0 such that the function ψ(s)/ϕ(s), 0 < s s0 , is non-increasing then there are a constant 1 C1 < ∞ and some α¯ > 0 such that x † − xα C1 ψ(α)w,

0 < α α. ¯

(21)

(b) If there is s0 > 0 such that the function ψ(s)/ϕ(s), 0 < s s0 , is non-decreasing then there are a constant 1 C2 < ∞ and some α¯ > 0 such that x † − xα C2 ϕ(α)w,

0 < α α. ¯

(22)

We postpone further discussion, in particular for Tikhonov regularization, to the end of section 3.3. 3.3. Convergence rates Having established both, bounds for the bias and for the noise propagation term, we obtain explicit error bounds (as functions of the noise level δ > 0) for the cases covered by proposition 4. As in usual regularization theory for general smoothness, see [7], we assign the strictly increasing (from zero to ∞) index function √ ψ (t) := tψ(t), 0 < t < ∞, (23) to any index function ψ. The convergence rates in the following result are the usual ones in regularization theory under bounded operators, and under the smoothness assumption x † = ψ(A∗ A)w. However, as was highlighted in the above discussion, the proof is not a straightforward consequence of the qualification, but needs to take into account the fact that we can replace unbounded index functions by appropriately chosen bounded ones. 11


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Theorem 2. Let gα be a regularization with known qualification ϕ. Suppose that the solution satisfies a source condition (3) given by the index function ψ with source element w ∈ D(ψ(H )). (a) If some estimate of the form (21) holds true, then we have the convergence rate † x − x δ = O ψ −1 (δ) as δ → 0, α(δ) ψ for an a priori parameter choice ψ (α(δ)) = δ. (b) If even some estimate of the form (22) is valid, then we have † x − x δ = O ϕ −1 (δ) as δ → 0, α(δ) ϕ for an a priori parameter choice ϕ (α(δ)) = δ. Proof. We first mention that by the properties of the functions ψ , ϕ , the parameters α(δ) exist. Now, the proof follows from the error decomposition (18), lemma 6 and proposition 4. In case (a) we have that † x − x δ C1 ψ(α)w + γ∗ √δ = (C1 w + γ∗ ) max ψ(α), √δ , α α α from which the first assertion follows. The proof of the second one is similar and we omit it. Example 2 (Tikhonov regularization). We shall review the outline in this section for Tikhonov regularization, i.e., when the solution is given by (12). This fits in the general approach by letting gα (t) := 1/(t + α), t > 0, α > 0, see remark 6. It is easily verified that the index function ϕ(s) = s, s > 0, is a qualification (with constant γ = 1) for Tikhonov regularization, both under bounded or unbounded operator A. Let us now discuss propositions 2 and 3 for Tikhonov regularization, and smoothness given in terms of monomials ψp (s) = s p , s > 0, for p > 0. Since the index function ϕ(s) = s, s > 0, is a qualification, the monomials ψp (s) satisfy the assumptions of proposition 2 whenever 0 < p 1, where the monotonicity of ψ(s)/s holds globally. If p > 1 then the quotient s p /s = s p−1 , s > 0, is globally increasing and tends to infinity as s → ∞. If the operator A is unbounded then (GC) fails to hold for p > 1. This means that we cannot apply proposition 3, and we cannot deduce a decay rate for the bias, even if the solution smoothness is high. However, we can replace the global monomial ψp by the function ψ0 which coincides with ψp on (0, 1], and is equal to 1 for s > 1. This modified index function is bounded and thus allows to apply proposition 4 and also theorem 2. We then obtain convergence rates of the order δ 2p/(2p+1) for 0 < p 1, and δ 2/3 for p > 1. This expresses the well-known effect of saturation for the method of Tikhonov regularization. More generally, suppose that gα is some regularization with qualification, say ϕ(s) := s p0 , s > 0 for some exponent p0 > 0. If the index function ψ is of power-type smoothness in a right neighborhood of s = 0, i.e., ψ(s) := s p , 0 < s s¯ , for some 0 < p < ∞, then the 2p corresponding bounds from theorem 2 are of the orders δ 2p+1 for 0 < p p0 . If p > p0 then 2p0

we obtain the rate δ 2p0 +1 as δ → 0. 12


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4. Two examples In the literature on linear ill-posed operator equations there are numerous examples of problems (1), where die operator A : D(A) ⊂ X → Y mapping between Hilbert spaces X and Y has zero and infinity as accumulation points of the spectrum. Frequently, for X and Y the L2-spaces of functions on unbounded domains Rk (k = 1, 2, . . .) are under consideration, which sometimes allows us to identify the corresponding multiplier function f according to lemmas 4 and 5 in an explicit manner. We mention two of such examples, analyzed in detail by U Tautenhahn and co-authors in [12, 13]. 4.1. Fractional differentiation In [13] the problem of fractional differentiation, i.e., of finding the derivative x = Dβ y of order 0 < β < 1 of a function y is analyzed. For applications of that problem in natural sciences and engineering we refer, e.g., to the monograph [3]. The problem can be written as an operator (fractional integration operator) equation (1) with X = Y = L2 (R), s x(t) 1 dt, s ∈ R. [Aβ x](s) =

(β) −∞ (s − t)1−β As mentioned by the authors of [13] the operator Aβ is densely defined, injective, normal and closed on its domain D(Aβ ) = {x ∈ L2 (R) : |s|−β [F x](s) ∈ L2 (R)},

where F is the Fourier transform operator. It can then be seen that the corresponding selfadjoint operator A∗β Aβ has the domain D(A∗β Aβ ) = {x ∈ L2 (R) : |s|−2β [F x](s) ∈ L2 (R)}.

Evidently, the corresponding multiplier function is fβ (s) = |s|−2β with R(fβ ) = (0, ∞) which indicates that H = A∗β Aβ has continuous spectrum. Hence (1) is ill-posed of type I (H non-compact). We briefly show how Sobolev smoothness of the solution x † , as discussed in example 1, translates to a source condition with respect to A∗β Aβ . Since both classes are related to multiplication by the Fourier transform, we see that we need to find a function ψ : (0, ∞) → (0, ∞) with ψ(|s|−2β ) = (1 + |s|2 )−p/2 ,

s ∈ R.

An easy calculation shows that this is achieved by 1/β p/2 t , t > 0. ψβ,p (t) := 1 + t 1/β The following result holds true, see also [13, proposition 3.1].

(24)

(25)

Lemma 8. The function ψβ,p from (25) is a bounded index function for which p . It is concave exactly if 0 < p 2β. ψβ,p (A∗β Aβ ) = G−1 1 This result allows us to apply theorem 2. Note that ψβ,p (t) t p/(2β) for all t > 0. If p 2β then this is concave, and theorem 2(a) applies and provides us with the convergence rate δ p/(p+β) if α is chosen as α ∼ δ 2β/(p+β) . Tikhonov regularization, when written in spectral domain is given by the Wiener filter, damping high frequency signals, as sβ [F y δ ](s), s ∈ R. [F y δ ](s) → 1 + αs 2β We have thus reproved the result for Tikhonov regularization in [13, theorem 4.1]. 13


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4.2. Cauchy problem for the Helmholtz equation The second example belongs to the field of inverse problems in partial differential equations. In [12] the authors study Cauchy problems for the Helmholtz equation u + k 2 u = 0,

(r, z) ∈ R2 × (0, d),

in a cylindrical domain with solution u(r,z) and wave number k > 0. Precisely, for fixed d and z ∈ [0, d) the function x = u(r, z) is to be determined from y = u(r, d), where u(r,z) satisfies the Helmholtz equation under homogeneous Neumann boundary conditions. We consider X = Y = L2 (R2 ) for fixed z ∈ (0, d). The inverse problem under consideration has applications in optoelectronics and in specific laser beam models and can be written as a linear operator equation Az u(r, z) = u(r, d),

(26)

where the operator Az is bounded for small wave numbers k(d − z) < and unbounded for large wave numbers k(d − z) > π2 . Fourier transform again yields the corresponding multiplier function, here [cosh((d − z) |t|2 − k 2 )]−2 if |t| k f (t) = , t = (t1 , t2 ). [cos((d − z) k 2 − |t|2 )]−2 if |t| k π 2

The authors of [12] demonstrate that the range R(f ) of f has an accumulation point zero in any case, and equation (26) is ill-posed. Specifically, the following results are proved. (a) In the case of large wave numbers they find that R(f ) = (0, ∞). (b) For small wave numbers k(d − z) < π/2, the operator Az is bounded and we have R(f ) = (0, [cos k(d − z)]−2 ]. (c) The initial condition {u ∈ D, u(·, 0) 1} translates to a source condition (3) given by the index function −1 d 1 1 , t > 0, ψz (t) := √ cosh acosh √ d −z t t see [12, proposition 3.1], i.e., x † ∈ Hψz := {x ∈ L2 (R2 ) : x = ψz (A∗z Az )v, v 1}.

(27)

A local analysis reveals that

√ z/(d−z) t as t → 0. 2 This shows two items. First, for each ε there is t¯ such that ψz (t) (1 + ε)ψ¯ z (t), 0 < t t¯. ψz (t) ∼ ψ˜ z (t) :=

(28)

Therefore, we may use the function (1 + ε)ψ¯ z (α) to bound the bias. Second, this bound is concave in the range 0 < z 23 d, i.e., when z is not close to the level d, and the degree of ill-posedness depends on the relative distance of z to the observation location d, see [12, section 3] for more details. (d) It follows from the bias bound related to (28) that the best possible accuracy expressed by the corresponding modulus of continuity on the set (27) for δ → 0 is given by z/d δ † δ sup x − R(y ) = (1 + O(1)), sup inf 2 x † ∈Hψz R Ax † −y δ δ where R : Y → X is any linear or nonlinear method of reconstruction. This slightly improves the bound given in [12, theorem 4.2]. 14


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The results of the previous section show that within the range 0 < z 23 d, Tikhonov regularization is appropriate for regularization, and it yields the error bound 3 δ z/d ¯ x † − xα∗ , 0 < δ δ, 2 2 with parameter choice α∗ := 4(δ 2 /4)(d−z)/d . This is close to the best possible accuracy, as mentioned in (d). Acknowledgments This work has been conducted during the Mini Special Semester on Inverse Problems, 18 May– 15 July 2009, organized by RICAM (Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Science) Linz, Austria. B Hofmann was supported by Deutsche Forschungsgemeinschaft (DFG) under grant HO1454/7-2. References [1] Engl H W, Hanke M and Neubauer A 1996 Regularization of Inverse Problems (Mathematics and its Applications vol 375) (Dordrecht: Kluwer) [2] Friedrichs K O 1980 Spectral Theory of Operators in Hilbert Space(Applied Mathematical Sciences vol 9) (New York: Springer) corrected reprint [3] Gorenflo R and Vessella S 1991 Abel Integral Equations. Analysis and Applications (Lecture Notes in Mathematics vol 1461) (Berlin: Springer) [4] Hofmann B and Fleischer G 1999 Stability rates for linear ill-posed problems with compact and non-compact operators Z. Anal. Anwendungen 18 267–86 [5] Hofmann B and Mathé P 2007 Analysis of profile functions for general linear regularization methods SIAM J. Numer. Anal. 45 1122–41 [6] Mathé P and Hofmann B 2008 How general are general source conditions? Inverse Problems 24 015009 [7] Mathé P and Pereverzev S V 2003 Geometry of linear ill-posed problems in variable Hilbert scales Inverse Problems 19 789–803 [8] Nashed M Z 1987 A new approach to classification and regularization of ill-posed operator equations Inverse and Ill-posed Problems (Sankt Wolfgang, 1986) (Notes Rep. Math. Sci. Engrg. vol 4) (Boston, MA: Academic) pp 53–75 [9] Pietsch A 1980 Operator Ideals (North-Holland Mathematical Library vol 20) (Amsterdam: North-Holland) (German Transl.) [10] Ramm A G 2008 On unbounded operators and applications Appl. Math. Lett. 21 377–82 [11] Reed M and Simon B 1980 Methods of Modern Mathematical Physics: I. Functional Analysis 2nd edn (New York: Academic) [12] Regińska T and Tautenhahn U 2009 Conditional stability estimates and regularization with applications to Cauchy problems for the Helmholtz equation Technical Report 706 Institute of Mathematics, Polish Academy of Sciences www.impan.pl/Preprints/ [13] Tautenhahn U and Gorenflo R 1999 On optimal regularization methods for fractional differentiation Z. Anal. Anwendungen 18 449–67 [14] Triebel H 1995 Interpolation Theory, Function Spaces, Differential Operators 2nd edn (Heidelberg: Johann Ambrosius Barth) [15] Va˘ınikko G M and Veretennikov A Yu 1986 Iteratsionnye Protsedury v Nekorrektnykh Zadachakh (Moscow: Nauka)

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